As we shall see, the statement follows as a consequence of the decomposability of \(H\) -action (Theorem REF ), Benoist–Quint [1]}, [2]} equidistribution results. We note however that we do not treat the question of equidistribution of trajectories of points \(x \in X \setminus X_\mathcal {C}\) . For such points, already in the level of the base space \(G/Q\) , the corresponding equidistribution question does not seem to be well-understood in all cases (cf.  [3]}).
